Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010; revised August 2010, revised November 2010 (essentially final version). (Besides many small changes, the first and second revised versions contain corrected entries in Tables I and II.

Lowering the Error Floor of LDPC Codes Using Cyclic Liftings

Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle $c$ of length $\ell(c)$ in the base graph such that the inverse image of $c$ in the lifted graph consists of only cycles of length strictly larger than $\ell(c)$. The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, encoder and decoder structure, and in some cases, the code rate. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel

Binary doubly-even self-dual codes of length 72 with large automorphism groups

We study binary linear codes constructed from fifty-four Hadamard 2-(71,35,17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss error-correcting capability of the obtained codes by majority logic decoding. Further, we describe a construction of a strongly regular graph with parameters (126,25,8,4) from a binary [35,8,4] code related to a derived 2-(35,17,16) design